Ascending and descending regions of a discrete Morse function

TL;DR

This paper introduces an algorithm for decomposing cellular complexes using discrete Morse functions, generalizing Morse-Smale decomposition to higher dimensions with proven correctness and practical performance analysis.

Contribution

It presents a new algorithm that generalizes Morse-Smale decomposition to any dimension, with proven correctness and applicability to cellular complexes.

Findings

Algorithm always produces a valid decomposition.

Regions become topological discs after finite subdivisions.

Demonstrated efficiency on several examples.

Abstract

We present an algorithm which produces a decomposition of a regular cellular complex with a discrete Morse function analogous to the Morse-Smale decomposition of a smooth manifold with respect to a smooth Morse function. The advantage of our algorithm compared to similar existing results is that it works, at least theoretically, in any dimension. Practically, there are dimensional restrictions due to the size of cellular complexes of higher dimensions, though. We prove that the algorithm is correct in the sense that it always produces a decomposition into descending and ascending regions of the critical cells in a finite number of steps, and that, after a finite number of subdivisions, all the regions are topological discs. The efficiency of the algorithm is discussed and its performance on several examples is demonstrated.